In the Name of Allah, Most Gracious, Most Merciful.
All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.
In this fatwa:
1- Qunut is supplication offered during prayer at a specific point while standing.
2- Qunut is recommended in Witr prayer after the ruku.
3- There is no sound report that Prophet Muhammad singled out Fajr prayer for qunut.
Answering your question, The Permanent Committee for Scholarly Research and Ifta’ in Saudi Arabia, states:
Qunut, according to the definition of the jurists, is the name of a duaa (supplication) offered during prayer at a specific point while standing.
If a calamity befalls the Muslims, it is recommended to say duaa al-qunut after standing up from ruku in the last rakah of each of the five daily obligatory prayers, until Allah relieves the Muslims of that calamity. (See Tasheeh al-Duaa by Bakr Abu Zayd, p. 460).
Qunut in Fajr Prayer
With regard to saying duaa al-qunut in Fajr prayer all the time, in all circumstances, there is no sound report that Prophet Muhammad (peace be upon him) singled out Fajr for qunut, or that he always recited it in Fajr prayer.
Rather, Prophet Muhammad said duaa al-qunut at times of calamity with words that were appropriate to the situation.
He said duaa al-qunut in Fajr and in other prayers, praying against Ra`l, Dhakwaan and `Usayyah for killing the Quran-readers whom he had sent to them in order to teach them their religion.
Prophet Muhammad offered duaa in Fajr prayer and other prayers for the weak and oppressed believers, that Allah would save them from their enemies. But he did not do that all the time.
The Rightly-Guided Caliphs after him followed the same practice. It is better for the imam to limit qunut to times of calamity, following the example of the Messenger of Allah.
Abu Malik al-Ashari said: “I said to my father, ‘O my father, you prayed behind the Messenger of Allah and behind Abu Bakr, Umar, Uthmaan and Ali. Did they used to say duaa al-qunut in Fajr?’ He said, ‘O my son, this is a newly-invented matter.’” (Abu Dawud; classed as sahih by al-Albani in al-Irwa, 435)
Almighty Allah knows best.